journal article Jan 01, 2023

Generalized differential identities on prime rings and algebras

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Abstract
<abstract><p>The goal of this study is to bring out the following conclusion. Let $ R $ be a noncommutative prime ring with $ 2(m+n)! $ torsion freeness and let $ m $ and $ n $ be fixed, non-negative integers and $ d, g $ be Jordan derivations on $ R $. If $ x^{m+n}d(x)+x^mg(x)x^n\in Z(R) $ or $ d(x)x^{m+n}+x^mg(x)x^n\in Z(R) $ or $ x^{n}d(x)x^{m}+x^mg(x)x^n\in Z(R) $ then $ d = g = 0 $ follows for every $ x\in R $.</p></abstract>
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References
19
[1]
S. Ali, N. A. K. Ali, A. M. Ansari, On $*$-differential identities in prime rings with involution, <i>Hacet. J. Math. Stat.</i>, <b>49</b> (2020), 708–715. https://doi.org/10.15672/hujms.588726 10.15672/hujms.588726
[2]
J. Cusack, Jordan derivations on rings, <i>P. Am. Math. Soc.</i>, <b>53</b> (1975), 321–324. 10.1090/s0002-9939-1975-0399182-5
[3]
B. Felzenswalb, Derivations in prime rings, <i>P. Am. Math. Soc.</i>, <b>84</b> (1982), 16–20. https://doi.org/10.1090/S0002-9939-1982-0633268-6 10.1090/s0002-9939-1982-0633268-6
[4]
B. L. M. Ferreira, W. Feng, Mixed $*$-Jordan-type derivations on $*$-algebras, <i>J. Algebra Appl.</i>, <b>22</b> (2023), 2350100. https://doi.org/10.1142/S0219498823501001 10.1142/s0219498823501001
[5]
B. L. M. Ferreira, H. Guzzo, R. N. Ferreira, F. Wei, Jordan derivations of alternative rings, <i>Commun. Algebra</i>, <b>48</b> (2020), 717–723. https://doi.org/10.1080/00927872.2019.1659285 10.1080/00927872.2019.1659285
[6]
I. N. Herstein, Jordan derivations of prime rings, <i>P. Am. Math. Soc.</i>, <b>8</b> (1957), 1104–1110. 10.1090/s0002-9939-1957-0095864-2
[7]
I. N. Herstein, A theorem on derivations of prime rings with involution, <i>Can. J. Math.</i>, <b>34</b> (1982), 356–369. https://doi.org/10.4153/CJM-1982-023-x 10.4153/cjm-1982-023-x
[8]
B. E. Johnson, A. M. Sinclair, Continuity of derivations and a problem of Kaplansky, <i>Am. J. Math.</i>, <b>90</b> (1968), 1067–1073. https://doi.org/10.2307/2373290 10.2307/2373290
[9]
C. Lanski, Differential identities in prime rings with involution, <i>T. Am. Math. Soc.</i>, <b>291</b> (1985), 765–787. https://doi.org/10.2307/2000109 10.2307/2000109
[10]
T. K. Lee, Semiprime rings with differential identities, <i>Bull. Inst. Math. Acad.</i>, <b>20</b> (1992), 27–38. Available from: <ext-link ext-link-type="uri" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="http://140.112.114.62/handle/246246/121932">http://140.112.114.62/handle/246246/121932</ext-link>.
[11]
M. Mathieu, G. J. Murphy, Derivations mapping into the radical, <i>Arch. Math.</i>, <b>57</b> (1991), 469–474. https://doi.org/10.1007/BF01246745 10.1007/bf01246745
[12]
M. Mathieu, V. Runde, Derivations mapping into the radical, II, <i>B. Lond. Math. Soc.</i>, <b>24</b> (1992), 485–487. https://doi.org/10.1112/blms/24.5.485 10.1112/blms/24.5.485
[13]
E. C. Posner, Derivations in prime rings, <i>P. Am. Math. Soc.</i>, <b>8</b> (1957), 1093–1100. 10.1090/s0002-9939-1957-0095863-0
[14]
A. M. Sinclair, Jordan homomorphisms and derivations on semisimple Banach algebra, <i>P. Am. Math. Soc.</i>, <b>24</b> (1970), 209–214. https://doi.org/10.1090/S0002-9939-1970-0250069-3 10.1090/s0002-9939-1970-0250069-3
[15]
A. M. Sinclair, <i>Automatic continuity of linear operators</i>, Cambridge University Press, 1976. <ext-link ext-link-type="uri" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="https://doi.org/10.1017/CBO9780511662355">https://doi.org/10.1017/CBO9780511662355</ext-link>
[16]
I. M. Singer, J. Werner, Derivations on commutative normed algebras, <i>Math. Ann.</i>, <b>129</b> (1955), 2–6. 10.1007/bf01362370
[17]
M. P. Thomas, The image of a derivation is contained in the radical, <i>Ann. Math.</i>, <b>128</b> (1988), 435–460. https://doi.org/10.2307/1971432 10.2307/1971432
[18]
M. P. Thomas, Primitive ideals and derivations on noncommutative Banach algebras, <i>Pac. J. Math.</i>, <b>159</b> (1993), 139–152. 10.2140/pjm.1993.159.139
[19]
F. Wei, Z. Xiao, Generalized derivations on (semi-) Prime rings and non commutative Banach algebras, <i>Rend. Semin. Mat. U. Pad.</i>, <b>122</b> (2009), 171–189. https://doi.org/10.4171/RSMUP/122-11 10.4171/rsmup/122-11
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Details
Published
Jan 01, 2023
Vol/Issue
8(10)
Pages
22758-22765
Cite This Article
Abu Zaid Ansari, Faiza Shujat, Ahlam Fallatah (2023). Generalized differential identities on prime rings and algebras. AIMS Mathematics, 8(10), 22758-22765. https://doi.org/10.3934/math.20231159